One number is 9 less than a second number. Find a pair of such number that their product is as small as possible.

Answer 1

Assign variables and write equations:

From the first sentence:

y = x - 9

The product is of course x*y. We need to write this with just one variable. we do that by replacing y with the expression from the 1st equation. we'll use P for the product (P(x) signifies that P is a function of x)

P(x) = x * (x - 9)

Expand to get P(x) = x² - 9x

To find a minimum or maximum, set the first derivative = to 0:

P' = 2x - 9 = 0

2x = 9

x = 4.5

Plot the function to verify that there is a minimum at x = 4.5

You can also solve this by first writing the general equation of a parabola:

y + a = k*(x + b)2 (For this case, k = 1)

to make x² - 9x a perfect square, add (-4.5)² which is 20.25. Adding this to both sides of the equation, we have

y + 20.25 = x² - 9x + 20.25

So, it is this parabola:

y + 20.25 = (x - 4.5)²

Which is an up-facing parabola with the vertex shifted down 20.25, and right 4.5

Answer 2

Let's call one of the numbers "x". Therefore, the other number is 9 less than x, or (x-9).

We want to minimize x(x-9) or x^2-9x.

If you graph the function y=x^2-9x, then you will see the minimum point (or the vertex) when x=4.5

So, if one number is 4.5 then the other number is -4.5. Their product is -20.25.